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arXiv:2611.00203

Exhaustive Construction of Generalized Parent Hamiltonians for Matrix Product States

He-Ran Wang, Lorenzo Gotta, Sanjay Moudgalya

quant-phcond-mat.stat-mechcond-mat.str-elmath-phSubmitted Oct 30, 2026• Announced Nov 3, 2026

Abstract

Characterizing local Hamiltonians that host interesting many-body states as eigenstates has been an important problem in quantum many-body physics. For matrix product states (MPS), there is a well-known construction of parent Hamiltonians that have the MPS as the frustration-free ground state. However, recent results on quantum many-body scars have shown that such a construction does not exhaust all possible Hamiltonians that have the MPS as an eigenstate. In this work, we develop an exhaustive construction of generalized parent Hamiltonians that have a given translation-invariant MPS as an eigenstate. These go beyond the conventional projector-based construction of frustration-free parent Hamiltonians, where each local Hermitian term has the MPS as an eigenstate, by allowing non-Hermitian telescoping terms that cancel out with periodic boundary conditions (PBC). We derive a local tensor equation that establishes a correspondence between inequivalent classes of generalized parent Hamiltonians, i.e., those that cannot be related using linear combinations with frustration-free Hamiltonians, and the MPS transfer matrix spectrum. We illustrate the connection to intrinsic physical properties, including non-Abelian symmetries and zero correlation length, and further extend this construction to block-injective MPS. Our framework hence provides a systematic bridge between generalized parent Hamiltonians and the local structure of quantum states.

Comments: 5+18 pages

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Characterizing local Hamiltonians that host interesting many-body states as eigenstates has been an important problem in quantum many-body physics. For matrix product states (MPS), there is a well-known construction of parent Hamiltonians that have the MPS as the frustration-free ground state. However, recent results on quantum many-body scars have shown that such a construction does not exhaust all possible Hamiltonians that have the MPS as an eigenstate. In this work, we develop an exhaustive construction of generalized parent Hamiltonians that have a given translation-invariant MPS as an eigenstate. These go beyond the conventional projector-based construction of frustration-free parent Hamiltonians, where each local Hermitian term has the MPS as an eigenstate, by allowing non-Hermitian telescoping terms that cancel out with periodic boundary conditions (PBC). We derive a local tensor equation that establishes a correspondence between inequivalent classes of generalized parent Hamiltonians, i.e., those that cannot be related using linear combinations with frustration-free Hamiltonians, and the MPS transfer matrix spectrum. We illustrate the connection to intrinsic physical properties, including non-Abelian symmetries and zero correlation length, and further extend this construction to block-injective MPS. Our framework hence provides a systematic bridge between generalized parent Hamiltonians and the local structure of quantum states.

Matrix product states (MPS) are quintessential examples of frustration-free gapped ground states of local interactions called parent Hamiltonians. In this work, we investigate parent Hamiltonians for a class of ergodic matrix product states (EMPS), which are MPS defined by site-dependent random tensors

\{X_j^{[k]}\}_{j=1}^D

which are homogeneously distributed at every site

k

in the spin chain. Here, the EMPS are not translation-invariant but rather statistically translation-invariant. Under a mild injectivity assumption, we show the thermodynamic limit of an EMPS is the unique frustration-free ground state of a parent Hamiltonian on the whole spin chain, which, depending on the statistical properties of the EMPS, may or may not be finite-range. In contrast to the translation-invariant regime, these Hamiltonians need not be gapped. Nevertheless, applying the martingale method while keeping track of local statistics gives conditions for a gap, in addition to pointing towards why there need not be a gap in general. We include examples of EMPS both with and without spectral gaps to illustrate our results.

The authors develop a general formalism to systematically construct parent Hamiltonians with quasiparticle excited states starting from a matrix product state (MPS) wave function. The resulting parent Hamiltonians are generically nonintegrable, and the quasiparticle excited states are thus examples of quantum many-body scars of the Hamiltonian. This method is used to recover well-known quantum scarred models, such as the AKLT spin chain, as well as to construct many new families of models, thus illustrating the close connection between parent Hamiltonians of MPS and quantum scarred models.

There are various research strategies used for non-Hermitian systems, which typically involve introducing non-Hermitian terms to preexisting Hermitian Hamiltonians. It can be challenging to directly design non-Hermitian many-body models that exhibit unique features not found in Hermitian systems. In this Letter, we propose a new method for constructing non-Hermitian many-body systems by generalizing the parent Hamiltonian method into non-Hermitian regimes. This allows us to build a local Hamiltonian using given matrix product states as its left and right ground states. We demonstrate this method by constructing a non-Hermitian spin-1 model from the asymmetric Affleck-Kennedy-Lieb-Tasaki state, which preserves both chiral order and symmetry-protected topological order. Our approach opens up a new paradigm for systematically constructing and studying non-Hermitian many-body systems, providing guiding principles for exploring new properties and phenomena in non-Hermitian physics.

Canonical forms are central to the analytical understanding of tensor network states, underpinning key results such as the complete classification of one-dimensional symmetry-protected topological phases within the matrix product state (MPS) framework. Yet, the established theory applies only to uniform MPS with periodic boundary conditions, leaving many physically relevant states beyond its reach. Here we introduce a generalized canonical form for uniform MPS with a boundary matrix, thus extending the analytical MPS framework to a more general setting of wider physical significance. This canonical form reveals that any such MPS can be represented as a block-invertible matrix product operator acting on a structured class of algebraic regular language states that capture its essential long-range and scale-invariant features. Our construction builds on new algebraic results of independent interest that characterize the span and algebra generated by non-semisimple sets of matrices, including a generalized quantum Wielandt's inequality that gives an explicit upper bound on the blocking length at which the fixed-length span stabilizes to an algebra. Together, these results establish a unified theoretical foundation for uniform MPS with boundaries, bridging the gap between periodic and arbitrary-boundary settings, and providing the basis for extending key analytical and classification results of matrix product states to a much broader class of states and operators.

The study of frustration-free Hamiltonians and their relation to finite bond-dimension matrix product states (MPS) has a long tradition. However, fractional quantum Hall (FQH) states do not quite fit into this theme since the known MPS representations of their ground states have infinite bond dimensions, which considerably obscures the relations between such MPS representations and the existence of frustration-free parent Hamiltonians. This is related to the fact that the latter necessarily are of infinite range in the orbital basis. Here, we present a theorem tailored to establishing the existence of frustration-free parent Hamiltonians in such a context. We explicitly demonstrate the utility of this theorem in the context of non-Abelian Moore-Read FQH states but argue the applicability of this theorem to transcend considerably beyond the realm of conformal-field-theory-derived MPSs or quasi-one-dimensional Hilbert spaces.

We study frustration-free Hamiltonians of fractional quantum Hall (FQH) states from the point of view of the matrix product state (MPS) representation of their ground and excited states. There is a wealth of solvable models relating to FQH physics, which, however, is mostly derived and analyzed from the vantage point of first-quantized ``analytic clustering properties.'' In contrast, one obtains long-ranged frustration-free lattice models when these Hamiltonians are studied in an orbital basis, which is the natural basis for the MPS representation of FQH states. The connection between MPS-like states and frustration-free parent Hamiltonians is the central guiding principle in the construction of solvable lattice models, but thus far, only for short-range Hamiltonians and MPSs of finite bond dimension. The situation in the FQH context is fundamentally different. Here we expose the direct link between the infinite-bond-dimension MPS structure of Laughlin--conformal field theory (CFT) states and their parent Hamiltonians. While focusing on the Laughlin state, generalizations to other CFT-MPSs will become transparent.